11.2: The Isomorphism Theorms
Although it is not evident at first, factor groups correspond exactly to homomorphic images, and we can use factor groups to study homomorphisms. We already know that with every group homomorphism \(\phi: G \rightarrow H\) we can associate a normal subgroup of \(G\text{,}\) \(\ker \phi\text{.}\) The converse is also true; that is, every normal subgroup of a group \(G\) gives rise to homomorphism of groups.
Let \(H\) be a normal subgroup of \(G\text{.}\) Define the natural or canonical homomorphism
\[ \phi : G \rightarrow G/H \nonumber \]
by
\[ \phi(g) = gH\text{.} \nonumber \]
This is indeed a homomorphism, since
\[ \phi( g_1 g_2 ) = g_1 g_2 H = g_1 H g_2 H = \phi( g_1) \phi( g_2 )\text{.} \nonumber \]
The kernel of this homomorphism is \(H\text{.}\) The following theorems describe the relationships between group homomorphisms, normal subgroups, and factor groups.
If \(\psi : G \rightarrow H\) is a group homomorphism with \(K =\ker \psi\text{,}\) then \(K\) is normal in \(G\text{.}\) Let \(\phi: G \rightarrow G/K\) be the canonical homomorphism. Then there exists a unique isomorphism \(\eta: G/K \rightarrow \psi(G)\) such that \(\psi = \eta \phi\text{.}\)
- Proof
-
We already know that \(K\) is normal in \(G\text{.}\) Define \(\eta: G/K \rightarrow \psi(G)\) by \(\eta(gK) = \psi(g)\text{.}\) We first show that \(\eta\) is a well-defined map. If \(g_1 K =g_2 K\text{,}\) then for some \(k \in K\text{,}\) \(g_1 k=g_2\text{;}\) consequently,
\[ \eta(g_1 K) = \psi(g_1) = \psi(g_1) \psi(k) = \psi(g_1k) = \psi(g_2) = \eta(g_2 K)\text{.} \nonumber \]
Thus, \(\eta\) does not depend on the choice of coset representatives and the map \(\eta: G/K \rightarrow \psi(G)\) is uniquely defined since \(\psi = \eta \phi\text{.}\) We must also show that \(\eta\) is a homomorphism. Indeed,
\begin{align*} \eta( g_1K g_2K ) & = \eta(g_1 g_2K)\\ & = \psi(g_1 g_2)\\ & = \psi(g_1) \psi(g_2)\\ & = \eta( g_1K) \eta( g_2K )\text{.} \end{align*}
Clearly, \(\eta\) is onto \(\psi( G)\text{.}\) To show that \(\eta\) is one-to-one, suppose that \(\eta(g_1 K) = \eta(g_2 K)\text{.}\) Then \(\psi(g_1) = \psi(g_2)\text{.}\) This implies that \(\psi( g_1^{-1} g_2 ) = e\text{,}\) or \(g_1^{-1} g_2\) is in the kernel of \(\psi\text{;}\) hence, \(g_1^{-1} g_2K = K\text{;}\) that is, \(g_1K =g_2K\text{.}\)
Mathematicians often use diagrams called commutative diagrams to describe such theorems. The following diagram “commutes” since \(\psi = \eta \phi\text{.}\)
Let \(G\) be a cyclic group with generator \(g\text{.}\) Define a map \(\phi : {\mathbb Z} \rightarrow G\) by \(n \mapsto g^n\text{.}\)
Solution
This map is a surjective homomorphism since
\[ \phi( m + n) = g^{m+n} = g^m g^n = \phi(m) \phi(n)\text{.} \nonumber \]
Clearly \(\phi\) is onto. If \(|g| = m\text{,}\) then \(g^m = e\text{.}\) Hence, \(\ker \phi = m {\mathbb Z}\) and \({\mathbb Z} / \ker \phi = {\mathbb Z} / m {\mathbb Z} \cong G\text{.}\) On the other hand, if the order of \(g\) is infinite, then \(\ker \phi = 0\) and \(\phi\) is an isomorphism of \(G\) and \({\mathbb Z}\text{.}\) Hence, two cyclic groups are isomorphic exactly when they have the same order. Up to isomorphism, the only cyclic groups are \({\mathbb Z}\) and \({\mathbb Z}_n\text{.}\)
Let \(H\) be a subgroup of a group \(G\) (not necessarily normal in \(G\) ) and \(N\) a normal subgroup of \(G\text{.}\) Then \(HN\) is a subgroup of \(G\text{,}\) \(H \cap N\) is a normal subgroup of \(H\text{,}\) and
\[ H / H \cap N \cong HN /N\text{.} \nonumber \]
- Proof
-
We will first show that \(HN = \{ hn : h \in H, n \in N \}\) is a subgroup of \(G\text{.}\) Suppose that \(h_1 n_1, h_2 n_2 \in HN\text{.}\) Since \(N\) is normal, \((h_2)^{-1} n_1 h_2 \in N\text{.}\) So
\[ (h_1 n_1)(h_2 n_2) = h_1 h_2 ( (h_2)^{-1} n_1 h_2 )n_2 \nonumber \]
is in \(HN\text{.}\) The inverse of \(hn \in HN\) is in \(HN\) since
\[ ( hn )^{-1} = n^{-1 } h^{-1} = h^{-1} (h n^{-1} h^{-1} )\text{.} \nonumber \]
Next, we prove that \(H \cap N\) is normal in \(H\text{.}\) Let \(h \in H\) and \(n \in H \cap N\text{.}\) Then \(h^{-1} n h \in H\) since each element is in \(H\text{.}\) Also, \(h^{-1} n h \in N\) since \(N\) is normal in \(G\text{;}\) therefore, \(h^{-1} n h \in H \cap N\text{.}\)
Now define a map \(\phi\) from \(H\) to \(HN / N\) by \(h \mapsto h N\text{.}\) The map \(\phi\) is onto, since any coset \(h n N = h N\) is the image of \(h\) in \(H\text{.}\) We also know that \(\phi\) is a homomorphism because
\[ \phi( h h') = h h' N = h N h' N = \phi( h ) \phi( h')\text{.} \nonumber \]
By the First Isomorphism Theorem, the image of \(\phi\) is isomorphic to \(H / \ker \phi\text{;}\) that is,
\[ HN/N = \phi(H) \cong H / \ker \phi\text{.} \nonumber \]
Since
\[ \ker \phi = \{ h \in H : h \in N \} = H \cap N\text{,} \nonumber \]
\(HN/N = \phi(H) \cong H / H \cap N\text{.}\)
Let \(N\) be a normal subgroup of a group \(G\text{.}\) Then \(H \mapsto H/N\) is a one-to-one correspondence between the set of subgroups \(H\) of \(G\) containing \(N\) and the set of subgroups of \(G/N\text{.}\) Furthermore, the normal subgroups of \(G\) containing \(N\) correspond to normal subgroups of \(G/N\text{.}\)
- Proof
-
Let \(H\) be a subgroup of \(G\) containing \(N\text{.}\) Since \(N\) is normal in \(H\text{,}\) \(H/N\) is a factor group. Let \(aN\) and \(bN\) be elements of \(H/N\text{.}\) Then \((aN)( b^{-1} N )= ab^{-1}N \in H/N\text{;}\) hence, \(H/N\) is a subgroup of \(G/N\text{.}\)
Let \(S\) be a subgroup of \(G/N\text{.}\) This subgroup is a set of cosets of \(N\text{.}\) If \(H= \{ g \in G : gN \in S \}\text{,}\) then for \(h_1, h_2 \in H\text{,}\) we have that \((h_1 N)( h_2 N )= h_1 h_2 N \in S\) and \(h_1^{-1} N \in S\text{.}\) Therefore, \(H\) must be a subgroup of \(G\text{.}\) Clearly, \(H\) contains \(N\text{.}\) Therefore, \(S = H / N\text{.}\) Consequently, the map \(H \mapsto H/N\) is onto.
Suppose that \(H_1\) and \(H_2\) are subgroups of \(G\) containing \(N\) such that \(H_1/N = H_2/N\text{.}\) If \(h_1 \in H_1\text{,}\) then \(h_1 N \in H_1/N\text{.}\) Hence, \(h_1 N = h_2 N \subset H_2\) for some \(h_2\) in \(H_2\text{.}\) However, since \(N\) is contained in \(H_2\text{,}\) we know that \(h_1 \in H_2\) or \(H_1 \subset H_2\text{.}\) Similarly, \(H_2 \subset H_1\text{.}\) Since \(H_1 = H_2\text{,}\) the map \(H \mapsto H/N\) is one-to-one.
Suppose that \(H\) is normal in \(G\) and \(N\) is a subgroup of \(H\text{.}\) Then it is easy to verify that the map \(G/N \rightarrow G/H\) defined by \(gN \mapsto gH\) is a homomorphism. The kernel of this homomorphism is \(H/N\text{,}\) which proves that \(H/N\) is normal in \(G/N\text{.}\)
Conversely, suppose that \(H/N\) is normal in \(G/N\text{.}\) The homomorphism given by
\[ G \rightarrow G/N \rightarrow \frac{G/N}{H/N} \nonumber \]
has kernel \(H\text{.}\) Hence, \(H\) must be normal in \(G\text{.}\)
Notice that in the course of the proof of Theorem \(11.13\), we have also proved the following theorem.
Let \(G\) be a group and \(N\) and \(H\) be normal subgroups of \(G\) with \(N \subset H\text{.}\) Then
\[ G/H \cong \frac{G/N}{H/N}\text{.} \nonumber \]
By the Third Isomorphism Theorem,
\[ {\mathbb Z} / m {\mathbb Z} \cong ({\mathbb Z}/ mn {\mathbb Z})/ (m {\mathbb Z}/ mn {\mathbb Z})\text{.} \nonumber \]
Solution
Since \(| {\mathbb Z} / mn {\mathbb Z} | = mn\) and \(|{\mathbb Z} / m{\mathbb Z}| = m\text{,}\) we have \(| m {\mathbb Z} / mn {\mathbb Z}| = n\text{.}\)


5: Two-Dimensional Data - Differences
All methods covered in this chapter based on the idea of statistical test and side-by-side comparison. If even there are methods which seemingly accept multiple samples (like ANOVA or analysis of tables), they internally do the same: compare two pooled variations, or expected and observed frequencies.
-
- 5.1: What is a statistical test?
- Philosophers postulated that science can never prove a theory, but only disprove it. If we collect 1000 facts that support a theory, it does not mean we have proved it—it is possible that the 1001st piece of evidence will disprove it. This is why in statistical testing we commonly use two hypotheses. The one we are trying to prove is called the alternative hypothesis (H₁). The other, default one, is called the null hypothesis (H₀).
-
- 5.2: Is there a difference? Comparing two samples
- Studying two samples, we use the same approach with two hypotheses. The typical null hypothesis is “there is no difference between these two samples”—in other words, they are both drawn from the same population. The alternative hypothesis is “there is a difference between these two samples”.


6.3: AESTHETICS
Aesthetics, the study of principles and appreciation of beauty, is linked to our thinking about and connections to art. During the eighteenth century in Europe, philosophers and other thinkers began to question the interrelationship of art, beauty, and pleasure. German philosopher Immanuel Kant characterized the appreciation of beauty as the “judgment of taste,” which is comprised of two parts: subjectivity and universality. Subjectivity, as the term suggests, is based on the feeling of pleasure or displeasure experienced by the individual viewer. Universality refers to views about art that are held in common, the “norm,” so to speak. Kant believed the beauty of art can only be appreciated when the viewer is “disinterested,” that is, when the viewer is deriving pleasure that is not based upon or produces desire. If the viewer’s subjective judgment is disinterested, then a universally valid measure of taste can be rendered. Only if the viewer can separate the appreciation of art from the desire for it, and is instead interested in art for its pure beauty, or aesthetics, can the viewer be said to have achieved the judgment of taste.
Writers, composers, and artists who were part of the Romantic movement that emerged in Europe at the end of the eighteenth century soon questioned Kant’s belief that aesthetics or the study of beauty in art, what he termed the judgment of taste, was both disinterested and universal. Turning away from categories and definitions based on rationality, Romanticism celebrated spontaneity, emotion, the individual, and the sublime : intellectual and imaginative sensations that defy measurement or explanation.
Romantic painter Eugène Delacroix (1798-1863, France) spent his lifetime seeking to express the extremes of human emotion and experience in his work based on history, literature, current events, and his own travels. With passages of brilliant color applied in thick, vigorous brush strokes, Delacroix depicted beauty, violence, tragedy, and ecstasy with equal passion, in waves of movement swiftly passing across his canvas. This quality can be seen in The Death of Sardanapalus where the shadowed figure of the Assyrian king surveys the scene of carnage taking place before him with dispassion. (Figure 6.1) Although historical accounts indicate that Sardanapalus did have all of his possessions destroyed, including his concubines and horses, rather than surrender them to his enemies, Delacroix largely relied on his own imagination for his frenzied interpretation and embellishment of the scene.
John Dewey, an American philosopher, psychologist, and educational reformer, in 1934 wrote Art and Experience . He described the aesthetic experience in ways that somewhat reflect the process Delacroix brought to his painting. Dewey stated, however, that although it begins with the art object, the experience of art extends far beyond that one element to produce an ongoing exchange between artist, viewer, and culture at large that culminates in an experience that is a “manifestation, a record and celebration of the life of a civilization.”1 The “sudden” pleasure one feels when engaging with a work of art or architecture is, in fact, the product of a long process of growth and engagement. For example, walking around and through a grand structure such as Reims Cathedral (1211-1275) in France, with its High Gothic façade, lavish sculptural decoration, extreme verticality, and expansive windows, is breathtakingly impressive because object (building) and experience have coalesced. (Figure 6.2) Further, we continue learning from the experience of observing art or beauty what, why, and when depends on how much we receive from the experience and from successive encounters.
The movement in thinking about aesthetics from Kant’s judgment of taste, with
its assumption of an intellectually based universality, to Dewey’s claim that the aesthetics of the work of art are found in the viewer’s experience of it, at that moment and over time, mirror substantial changes that have taken place in all aspects of scientific and intellectual thought over the past three centuries. What we can learn from their theories is that we can examine ideas about “fine arts,” “beauty,” and “aesthetics” and perhaps come up with similar definitions conveying ideas of pleasure, temporary enlightenment, and human experience but, we may not.
For example, Miami-based artist Jona Cerwinske (USA), began his career making graffiti art and street murals and considers any surface a ground for art. In 2007, he covered a Lamborghini car with an intertwined network of organic shapes and geometric lines. (Lamborghini Art, Jona Cerwinske: www.dubmagazine.com/home/cars...ske-exotic-art) This work of art could be described as an example of disinterested contemplation: you look at the Lam- borghini and contemplate the beauty and elegance of the car and its design. In this way, the car’s aesthetic appeal stems from admiration of the object and the delight it gives; it is a judgment of taste. Conversely, it could be described as an aesthetic experience: looking at the Lamborghini pro- duces a response of pleasure, perhaps at its beauty, its place in the history of fine motor cars, or the thought of owning and driving such a prestigious and fast vehicle. In this case, appreciation of beauty is both a broadly intellectual as well as an individual emotional response.


3.3: Introduction to Assessing the Fit of a Line
What you’ll learn to do: Use residuals, standard error, and r 2 to assess the fit of a linear model.
Graphing the regression line with the scatterplot gives a visual depiction of how well the regression line fits the data. To further hone in on assessing the fit of our regression line to the data, in this section we present:
- Residual plots.
- The correlation coefficient r gives us a numerical way to measure this fit.
- Interpreting the square of the correlation coefficient r 2 .
- Interpreting the standard error s e .
Contributors and Attributions
CC licensed content, Shared previously
- Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution


13: Fundamental Equilibrium Concepts
In this chapter, you will learn how to predict the position of the balance and the yield of a product of a reaction under specific conditions, how to change a reaction's conditions to increase or reduce yield, and how to evaluate an equilibrium system's reaction to disturbances.
-
- 13.1: Introduction
- Reactions can occur in both directions simultaneously (reactants to products and products to reactants) and eventually reach a state of balance. Reactions can occur in both directions simultaneously (reactants to products and products to reactants) and eventually reach a state of balance.
-
- 13.2: Chemical Equilibria
- A reaction is at equilibrium when the amounts of reactants or products no longer change. Chemical equilibrium is a dynamic process, meaning the rate of formation of products by the forward reaction is equal to the rate at which the products re-form reactants by the reverse reaction.
-
- 13.3: Equilibrium Constants
- For any reaction that is at equilibrium, the reaction quotient Q is equal to the equilibrium constant K for the reaction. If a reactant or product is a pure solid, a pure liquid, or the solvent in a dilute solution, the concentration of this component does not appear in the expression for the equilibrium constant. At equilibrium, the values of the concentrations of the reactants and products are constant and the reaction quotient will always equal K.
-
- 13.4: Shifting Equilibria - Le Chatelier’s Principle
- Systems at equilibrium can be disturbed by changes to temperature, concentration, and, in some cases, volume and pressure; volume and pressure changes will disturb equilibrium if the number of moles of gas is different on the reactant and product sides of the reaction. The system's response to these disturbances is described by Le Châtelier's principle: The system will respond in a way that counteracts the disturbance. Not all changes to the system result in a disturbance of the equilibrium.
-
- 13.5: Equilibrium Calculations
- The ratios of the rate of change in concentrations of a reaction are equal to the ratios of the coefficients in the balanced chemical equation. The sign of the coefficient of X is positive when the concentration increases and negative when it decreases. We learned to approach three basic types of equilibrium problems. When given the concentrations of the reactants and products at equilibrium, we can solve for the equilibrium constant.
-
- 13.9: Exercises
- These are homework exercises to accompany the Textmap created for "Chemistry" by OpenStax. Complementary General Chemistry question banks can be found for other Textmaps and can be accessed here. In addition to these publicly available questions, access to private problems bank for use in exams and homework is available to faculty only on an individual basis; please contact Delmar Larsen for an account with access permission.


1.3: Inclusion
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When we talk about inclusive early childhood education, we are talking about programs that are designed for children and families from a wide range of backgrounds. A program can serve diverse families, but not be truly inclusive; you can have families from different backgrounds in your program, but do they feel welcome? Are you forcing them to adapt to a program that isn’t designed for them or that doesn’t take into consideration their social, cultural, and/or racial contexts? Designing an inclusive program translates to everything from physical space to how you interact with children and families. [14]
Inclusion embodies the values, policies, and practices that support the right of every child and their family, regardless of ability, to participate in a broad range of activities and contexts as full members of families, communities, and society. The desired results of inclusive experiences for children with and without disabilities and their families include a sense of belonging and membership, positive social relationships and friendships, and development and learning to reach their full potential. The defining features of inclusion that can be used to identify high quality early childhood programs and services are:
Children with disabilities and their families continue to face significant barriers to accessing inclusive high-quality early childhood programs and too many preschool children with disabilities are only offered the option of receiving special education services in settings separate from their peers without disabilities. [17]
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Think About It… Describe your understanding of how diversity, equity, and inclusion are related. |


3.14: Types of Plans and Common Planning Tools
- Differentiate between the uses of long-term plans, short-term plans, and operational plans.
- Differentiate between standing plans and single-use plans.
- Explain how policies, procedures, and regulations impact operational plans.
- Explain the role of budgets in the planning process.
- Differentiate between forecasting, scenario planning, and contingency planning.
- Explain the use of “management by objectives” (MBO), SMART goals, and benchmarking in planning.
You can view the transcript for “Long term and short term planning animated” here (opens in new window) .
Watch the short animated video for a brief overview of the importance of long-term and short-term planning.
Long-term and Short-term Plans
When you decided to attend college, you had a long-term plan in mind.  Or, you may have committed two or three years to become a nurse, a medical technician, or an electrician. Your long-term goal was necessary to make sure that your daily activities would help you achieve your desired outcome. You could have just enrolled in a school and taken classes that looked interesting, but then where would you be in four years? You most likely would not have taken the courses required to qualify you for the job you want. An organization, especially a business, is not so different. It also needs a long-term plan to make sure that the daily activities of its employees are contributing to the mission and value statements of the organization.
A long-term plan is crucial to the ultimate success of the organization. A long-term plan for many businesses, such as construction, hospitality, or manufacturing, generally extends four to five years into the future. For other faster-changing industries, especially technology companies, a long-term plan may only look two or three years into the future. After that, it becomes too difficult to predict the future with any degree of certainty.
Top management is responsible for the development of the long-term plan. It is up to the CEO to make sure that changing conditions (both external and internal) are reflected in the organization’s long-term plan. The larger and more complex the organization, the larger and more complex the long-term plan will be to include all of the individual departments and functions.
Short-term plans generally allocate resources for a year or less. They may also be referred to as operational plans because they are concerned with daily activities and standard business operations. Like long-term plans, short-term plans must be monitored and updated, and this is the role of middle- and first-level management. Different managerial levels have responsibility for implementing different types of short-term plans. For example, a department manager may be comfortable implementing an operational plan for the entire year for her department. A marketing manager may direct a three- to four-month plan that involves the introduction of a new product line. A team leader may only be comfortable planning and implementing very specific activities over the period of a month.
Operational Plans: Standing Plans and Single-Use Plans
An operational plan describes the specific goals and objectives and milestones set by an organization during a specific period. ( Objectives are specific tasks undertaken to meet broader goals. A goal may be to increase product sales by 3 percent; an objective may be to hire two additional sales agents.) It will allocate the tangible resources (labor, equipment, space) and authorize the financing necessary to meet the objectives of the plan. There are two types of operational plans: standing plans and single-use plans.
- Standing plans are plans designed to be used again and again. Examples include policies, procedures, and regulations. The advantage of standing plans is that they foster unity and fairness within an organization and help to support stated organizational values. Managers don’t have to make unique decisions already addressed by various organizational policies. Standing plans also save time because managers know in advance how to address common situations. Finally, standing plans aid in the delegation of work, because employees are already familiar with the procedures and regulations followed by the organization.
- Single-use plans refer to plans that address a one-time project or event. The length of the plans varies, but the most common types are budgets and project schedules. The obvious advantage of a single-use plan is that it can be very specific in how it addresses the needs of a particular situation.
Policies, Procedures, and Regulations
As stated above, the most common examples of standing use plans are policies, procedures, and regulations. These plans are usually published and handed out to new hires or posted on the organization’s employee website for easy reference.
- Policies provide broad guidelines for the smooth operation of the organization. They cover things like hiring and firing, performance appraisals, promotions, and discipline. For example, a company may have a policy to encourage recycling in the workplace or a policy that prohibits personal cell phone use in manufacturing areas.
- Procedures are steps to be followed in established and repeated operations. Procedures should reflect the policies of the company and support the organization’s long-term goals. Procedures may also detail steps that should be followed to ensure employees are disciplined in a fair and unbiased manner. For example, if employees feel that other employees interacted with them in an inappropriate manner, then they should follow the procedure for bringing this to management’s attention. Or, the organization may establish procedures for what to do in cases of emergencies, such as a fire or toxic spill.
- Regulations refer to what is allowable and what is strictly prohibited in an organization. In other words, a regulation is a kind of rule that addresses general situations. In many hospitals and laboratories, for example, there are safety regulations against wearing open-toed shoes or shoes with slippery soles. State and federal governments frequently issue regulations for industries that impact public safety.
The Role of Budgets in the Planning Process
Refer back to Figure 1 and locate the box labeled “Budgets.” Notice that budgets are examples of single-use, short-term plans. An organization’s budget is a document that details the financial and physical resources allocated to a project or department. They are single-use plans because they are specific to a particular period or event. For example, departments may have a hiring budget that allocates a certain number of positions and a total salary value for a calendar year. Next year, that budget may be the same or it may change, depending upon conditions in the organization. But it cannot be assumed that the budget will stay the same. Zero-based budgets look at each budget as if it were brand new and require managers to justify each of the budgeted items. This process ensures that budgets are closely tied to the latest organizational goals.
Managers deal with a variety of budget types:
- Financial budgets include balance sheets, income/expense statements, and statements of cash flow.
- Operating budgets project revenue against expenditures.
- Nonmonetary budgets allocate resources such as labor, workspace, and equipment use.
- Fixed budgets are budgets that do not change with increased or decreased activities, such as sales revenue. They are also called static budgets.
- Flexible budgets will vary with the level of activity (grow or be reduced according to changing conditions).
Budgets are a very important planning tool, and organizations take their budgeting process very seriously. Some managers spend most of their time making sure that the expenses and projects they control do not exceed authorized spending limits. To routinely “go over budget” is a sign of a poor planning—and planning is one of the basic management functions. In some cases, to routinely come in under budget is also viewed negatively, because with more accurate budgeting those committed resources could have been allocated to other projects. Often, projects compete for limited resources so the best budget is the one that most closely projects actual expenses and revenue.
Forecasting, Scenario Planning, and Contingency Planning
Forecasting is simply making a prediction about the future. Anyone can make a forecast—the trick is to be right or close enough so that important planning decisions can be based on the forecast. Some “botched” forecasts by business leaders follow:
“This telephone has too many shortcomings to be seriously considered as a means of communication.” – President of Western Union, 1876
“There is a world market for maybe five computers.” – Chairman of IBM, 1943
“Television won’t be able to hold on to any market it captures after the first six months. People will soon get tired of staring at a plywood box every night.” – Darryl Zanuck, president of 20th Century Fox, 1946
“There is no chance that the iPhone is going to get any significant market share.” – Microsoft CEO Steve Ballmer, 2012
Scientific forecasting is using mathematical models, historical data, and statistical analysis to make predictions about what will happen in the future. Businesses use short-term forecasting all the time when creating budgets and anticipating expenses. Mostly, these forecasts are based on what they sold and what they paid providers in the recent past. Long-range forecasting requires both quantitative numerical data and qualitative data based on expert opinions and insights. Often, organizations will create a number of long-range forecasts based on “best-case” and “worst-case” scenarios. They will then make plans on how they would respond to each situation and, as time goes on, they will update and adapt the long-term plan.
One other important type of planning is the contingency plan . A contingency plan describes what will happen in a possible—but not expected—situation. Usually, contingency plans are designed to handle emergency situations. For example, airports have contingency plans for plane crashes on takeoffs or landings, and popular tourist attractions have begun developing contingency plans in case of terrorist threats.
An example of the critical importance of contingency planning involves the Deepwater Horizon oil spill in the Gulf of Mexico in 2010. Eleven people lost their lives and seventeen were badly injured when an explosion on an oil rig released almost five million barrels of oil into the Gulf of Mexico. It was the worst marine oil spill in history, and its effects were even more devastating because BP Oil did not have contingency plans in place for that kind of disaster. The spill went on for months while BP and its partners tried to figure out how to shut off the oil’s source. Even though BP spent $62 billion on the response and cleanup activities, there was extensive damage to marine and wildlife habitats and fishing and tourism industries. Getting employees involved in planning may help prevent tragedies similar to this one.
Management By Objectives (MBO) and SMART Goals
Management by objectives , or MBO for short, is a tool that can be used to improve the performance of an organization by creating clearly defined objectives agreed upon by management and by the employees. Peter Drucker, a prolific author and a leader in management theory, coined the phrase “management by objectives” in 1954. The intent of MBO is to improve employee motivation and organizational communication by focusing on aligning individual goals to corporate objectives. In MBO, a manager and an employee do the following:
- jointly set goals and objectives for a period.
- together plan tasks that the employee performs with the support of management.
- agree on the standards for evaluating performance of the task.
- regularly meet to review progress.
MBO must be a top-down management tool, because organizational goals are cascaded down to create the various operational levels. Drucker showed that as long as employee goals support short-term and long-term organizational objectives, MBO will help move the company forward. Critics, however, charge that managers using the approach focus more on creating goals than on helping the employee achieve them.
SMART goals are a technique often paired with MBO. SMART stands for specific, measurable, achievable, realistic, and time-bound. The SMART goal paired well with MBO theory by
- Providing incentives to employees by rewarding them when they meet key goals.
- Empowering employees by allowing them to set their own objectives for achieving their individual goal.
- Communicating honestly about what went well and what did not, and focusing on developing the missing skills.
The chart that follows summarizes the most important characteristics of each part of a SMART goal.
| SPECIFIC | Objectives must be specific enough to avoid confusion. They should identify what the objective is in terms of action to be done. |
|---|---|
| MEASURABLE | The objective should be numerical and quantifiable. Avoid term such as some, most, many, and enough. |
| ACHIEVABLE | An objective should be able to be met with reasonable effort. They should not be the source of undue physical or emotional stress. |
| RELEVANT | The objective should contribute in some way to the success of the organization as well as the development of the employee. |
| TIME-BOUND | Objectives should come with firm dates—not “soon” or “in a timely manner.” |
For example, let’s say you set a goal to become a recognized department expert in a subject relevant to advancement within the organization. How could you turn this into a SMART goal?
- Specific: I will learn about the liabilities of six major nonprofit organizations.
- Measurable: I will make presentations to the advertising, grant writing, and donor/client committees.
- Achievable: I will interview one nonprofit organization every week for six weeks.
- Relevant: This expertise will fill a current knowledge gap in the new client department.
- Time-bound: I will fulfill this goal before my next scheduled annual performance evaluation.
Benchmarking
The last planning tool we’ll discuss in this section is benchmarking. You may think that your organization has an excellent long-term plan and effective short-term plans, but how do you really know? Even if your company is showing growth, is it growing as fast as your competitor? A benchmark is a standard used for comparison purposes. Benchmarking is looking at performance levels outside of your organization, or sometimes across departments or divisions inside your organization, to evaluate your own performance. You can benchmark using several different criteria:
- Industry: Let’s say you produce technology widgets. Benchmarking can answer questions about how your company is doing in comparison to other tech widget makers. This approach is a type of competitive benchmarking.
- Geography: Your state is showing a lot of economic growth. You can use benchmarking to determine if your company is sharing in that wealth or underperforming compared to the regional economy.
 Benchmarking can help answer questions about whether the economic climate is friendlier to big business than it is to small business, or whether nonprofits are failing whereas for-profits are succeeding.
- Processes: You can use benchmarking to determine what processes other firms are using that are helping or hurting them. Are there lessons to be learned from them? This is also called strategic benchmarking or process benchmarking.
- Innovation: Benchmarking can help you discover what partners or techniques your competitors are using that are missing in your organization. Are there functions in your products or programs that should be eliminated and others that could be added? Functional benchmarking is key in technology-related organizations.
Internal benchmarking means comparing a department’s performance with another department in your company or branch within the same larger organization. The important thing about benchmarking is that it gives you a standard against which to compare your progress.
Key Points
Planning tools are designed to help you determine goals, guide behaviors within the organization, and help you evaluate your performance against external benchmarks. Plans are essential, but good managers know to be flexible when conditions demand.
Contributors and Attributions
- Types of Plans and Planning Tools. Authored by : John/Lynn Bruton and Lumen Learning. License : CC BY: Attribution
- Image: Organizational Plan Hierarchy. Authored by : Lumen Learning. License : CC BY: Attribution
- Image: PsychicBoston. Authored by : John Stephen Dwyer. Located at : https://commons.wikimedia.org/wiki/File:PsychicBoston.jpg . License : CC BY-SA: Attribution-ShareAlike
- Long term and short term planning animated. Authored by : Artem Nedoshepa. Located at : https://www.youtube.com/watch?v=De0HyiqRXIU . License : All Rights Reserved


12.E: Quantum Mechanics (Exercise)
Conceptual Questions
13.1: Blackbody Radiation
1. Give an example of a physical entity that is quantized. State specifically what the entity is and what the limits are on its values.
2. Give an example of a physical entity that is not quantized, in that it is continuous and may have a continuous range of values.
3. What aspect of the blackbody spectrum forced Planck to propose quantization of energy levels in the thermal oscillators?
4. Why don’t we notice quantization in everyday events?
13.2: The Photoelectric Effect
5. Is visible light the only type of EM radiation that can cause the photoelectric effect?
6. Which aspects of the photoelectric effect cannot be explained without photons? Which can be explained without photons? Are the latter inconsistent with the existence of photons?
7. Is the photoelectric effect a direct consequence of the wave character of EM radiation or of the particle character of EM radiation? Explain briefly.
8. Insulators (nonmetals) have a higher BE than metals, and it is more difficult for photons to eject electrons from insulators. Discuss how this relates to the free charges in metals that make them good conductors.
9. If you pick up and shake a piece of metal that has electrons in it free to move as a current, no electrons fall out. Yet if you heat the metal, electrons can be boiled off. Explain both of these facts as they relate to the amount and distribution of energy involved with shaking the object as compared with heating it.
13.3: The Wave Nature of Matter
10. How does the interference of water waves differ from the interference of electrons? How are they analogous?
11. Describe one type of evidence for the wave nature of matter.
12. Describe one type of evidence for the particle nature of EM radiation.
13.4: Uncertainty Principle
13. What is the Heisenberg uncertainty principle? Does it place limits on what can be known?
13.5: Discovery of the Atomic Nucleus
14. What two pieces of evidence allowed the first calculation of \(m_{e}\), the mass of the electron?
(a) The ratios \(q_{e} / m_{e}\) and \(q_{p} / m_{p}\).
(b) The values of \(q_{e}\) and \(E_{B}\).
(c) The ratio \(q_{e} / m_{e}\) and \(q_{e}\).
Justify your response.
15. How do the allowed orbits for electrons in atoms differ from the allowed orbits for planets around the sun? Explain how the correspondence principle applies here.
13.6: Bohr’s Theory of the Hydrogen Atom
16. How do the allowed orbits for electrons in atoms differ from the allowed orbits for planets around the sun? Explain how the correspondence principle applies here.
17. Explain how Bohr’s rule for the quantization of electron orbital angular momentum differs from the actual rule.
18. What is a hydrogen-like atom, and how are the energies and radii of its electron orbits related to those in hydrogen?
13.7: The Wave Nature of Matter Causes Quantization
19. How is the de Broglie wavelength of electrons related to the quantization of their orbits in atoms and molecules?
Problems & Exercises
13.1: Blackbody Radiation
20. A LiBr molecule oscillates with a frequency of \(1.7 \times 10^{13} \mathrm{~Hz}\)
(a) What is the difference in energy in eV between allowed oscillator states?
(b) What is the approximate value of \(n\) for a state having an energy of 1.0 eV?
Solution
(a) 0.070 eV
(b) 14
21. The difference in energy between allowed oscillator states in HBr molecules is 0.330 eV. What is the oscillation frequency of this molecule?
22. A physicist is watching a 15-kg orangutan at a zoo swing lazily in a tire at the end of a rope. He (the physicist) notices that each oscillation takes 3.00 s and hypothesizes that the energy is quantized.
(a) What is the difference in energy in joules between allowed oscillator states?
(b) What is the value of \(n\) for a state where the energy is 5.00 J?
(c) Can the quantization be observed?
13.2: The Photoelectric Effect
23. What is the longest-wavelength EM radiation that can eject a photoelectron from silver, given that the binding energy is 4.73 eV? Is this in the visible range?
Solution
263 nm
24. Find the longest-wavelength photon that can eject an electron from potassium, given that the binding energy is 2.24 eV. Is this visible EM radiation?
25. What is the binding energy in eV of electrons in magnesium, if the longest-wavelength photon that can eject electrons is 337 nm?
Solution
3.69 eV
26. Calculate the binding energy in eV of electrons in aluminum, if the longest-wavelength photon that can eject them is 304 nm.
27. What is the maximum kinetic energy in eV of electrons ejected from sodium metal by 450-nm EM radiation, given that the binding energy is 2.28 eV?
Solution
0.483 eV
28. UV radiation having a wavelength of 120 nm falls on gold metal, to which electrons are bound by 4.82 eV. What is the maximum kinetic energy of the ejected photoelectrons?
29. Violet light of wavelength 400 nm ejects electrons with a maximum kinetic energy of 0.860 eV from sodium metal. What is the binding energy of electrons to sodium metal?
Solution
2.25 eV
30. UV radiation having a 300-nm wavelength falls on uranium metal, ejecting 0.500-eV electrons. What is the binding energy of electrons to uranium metal?
31. What is the wavelength of EM radiation that ejects 2.00-eV electrons from calcium metal, given that the binding energy is 2.71 eV? What type of EM radiation is this?
Solution
(a) 264 nm
(b) Ultraviolet
32. Find the wavelength of photons that eject 0.100-eV electrons from potassium, given that the binding energy is 2.24 eV. Are these photons visible?
33. A laser with a power output of 2.00 mW at a wavelength of 400 nm is projected onto calcium metal.
(a) How many electrons per second are ejected?
(b) What power is carried away by the electrons, given that the binding energy is 2.71 eV?
Solution
(a) \(4.02 \times 10^{15} / \mathrm{s}\)
(b) 0.256 mW
34. (a) Calculate the number of photoelectrons per second ejected from a 1.00-mm 2 area of sodium metal by 500-nm EM radiation having an intensity of \(1.30 \mathrm{~kW} / \mathrm{m}^{2}\) (the intensity of sunlight above the Earth’s atmosphere).
(b) Given that the binding energy is 2.28 eV, what power is carried away by the electrons? (c) The electrons carry away less power than brought in by the photons. Where does the other power go? How can it be recovered?
Unreasonable Results
35. Red light having a wavelength of 700 nm is projected onto magnesium metal to which electrons are bound by 3.68 eV.
(a) Use \(\mathrm{KE}_{e}=h f-\mathrm{BE}\) to calculate the kinetic energy of the ejected electrons.
(b) What is unreasonable about this result?
(c) Which assumptions are unreasonable or inconsistent?
Solution
(a) \(-1.90 ~\mathrm{eV}\)
(b) Negative kinetic energy
(c) That the electrons would be knocked free.
Unreasonable Results
36. (a) What is the binding energy of electrons to a material from which 4.00-eV electrons are ejected by 400-nm EM radiation?
(b) What is unreasonable about this result?
(c) Which assumptions are unreasonable or inconsistent?
13.3: The Wave Nature of Matter
37. At what velocity will an electron have a wavelength of 1.00 m?
Solution
\(7.28 \times 10^{-4} \mathrm{~m}\)
38. What is the wavelength of an electron moving at 3.00% of the speed of light?
39. At what velocity does a proton have a 6.00-fm wavelength (about the size of a nucleus)? Assume the proton is nonrelativistic. (1 femtometer = \(10^{-15} \mathrm{~m}\))
Solution
\(6.62 \times 10^{7} \mathrm{~m} / \mathrm{s}\)
40. What is the velocity of a 0.400-kg billiard ball if its wavelength is 7.50 cm (large enough for it to interfere with other billiard balls)?
41. Find the wavelength of a proton moving at 1.00% of the speed of light.
Solution
\(1.32 \times 10^{-13} \mathrm{~m}\)
42. Experiments are performed with ultracold neutrons having velocities as small as 1.00 m/s. (a) What is the wavelength of such a neutron? (b) What is its kinetic energy in eV?
43. (a) Find the velocity of a neutron that has a 6.00-fm wavelength (about the size of a nucleus). Assume the neutron is nonrelativistic.
(b) What is the neutron’s kinetic energy in MeV?
Solution
(a) \(6.62 \times 10^{7} \mathrm{~m} / \mathrm{s}\)
(b) \(22.9 \mathrm{~MeV}\)
44. What is the wavelength of an electron accelerated through a 30.0-kV potential, as in a TV tube?
45. What is the kinetic energy of an electron in a TEM having a 0.0100-nm wavelength?
Solution
15.1 keV
46. (a) Calculate the velocity of an electron that has a wavelength of \(1.00 ~\mu \mathrm{m}\)
(b) Through what voltage must the electron be accelerated to have this velocity?
47. The velocity of a proton emerging from a Van de Graaff accelerator is 25.0% of the speed of light.
(a) What is the proton’s wavelength?
(b) What is its kinetic energy, assuming it is nonrelativistic?
(c) What was the equivalent voltage through which it was accelerated?
Solution
(a) 5.29 fm
(b) \(4.70 \times 10^{-12} \mathrm{~J}\)
(c) 29.4 MV
48. The kinetic energy of an electron accelerated in an X-ray tube is 100 keV. Assuming it is nonrelativistic, what is its wavelength?
Unreasonable Results
49. (a) Assuming it is nonrelativistic, calculate the velocity of an electron with a 0.100-fm wavelength (small enough to detect details of a nucleus).
(b) What is unreasonable about this result?
(c) Which assumptions are unreasonable or inconsistent?
Solution
(a) \(7.28 \times 10^{12} \mathrm{~m} / \mathrm{s}\)
(b) This is thousands of times the speed of light (an impossibility).
(c) The assumption that the electron is non-relativistic is unreasonable at this wavelength.
13.4: Uncertainty Principle
50. (a) If the position of an electron in a membrane is measured to an accuracy of \(1.00 ~\mu \mathrm{m}\), what is the electron’s minimum uncertainty in velocity?
(b) If the electron has this velocity, what is its kinetic energy in eV?
(c) What are the implications of this energy, comparing it to typical molecular binding energies?
Solution
(a) 57.9 m/s
(b) \(9.55 \times 10^{-9} ~\mathrm{eV}\)
(c) Typical molecular binding energies range from about 1eV to 10 eV, therefore the result in part (b) is approximately 9 orders of magnitude smaller than typical molecular binding energies.
51. (a) If the position of a chlorine ion in a membrane is measured to an accuracy of \(1.00 ~\mu \mathrm{m}\), what is its minimum uncertainty in velocity, given its mass is \(5.86 \times 10^{-26} \mathrm{~kg}\)?
(b) If the ion has this velocity, what is its kinetic energy in eV, and how does this compare with typical molecular binding energies?
52. Suppose the velocity of an electron in an atom is known to an accuracy of \(2.0 \times 10^{3} \mathrm{~m} / \mathrm{s}\) (reasonably accurate compared with orbital velocities). What is the electron’s minimum uncertainty in position, and how does this compare with the approximate 0.1-nm size of the atom?
Solution
29 nm,
290 times greater
53. The velocity of a proton in an accelerator is known to an accuracy of 0.250% of the speed of light. (This could be small compared with its velocity.) What is the smallest possible uncertainty in its position?
54. A relatively long-lived excited state of an atom has a lifetime of 3.00 ms. What is the minimum uncertainty in its energy?
Solution
\(1.10 \times 10^{-13} ~\mathrm{eV}\)
55. (a) The lifetime of a highly unstable nucleus is \(10^{-20} \mathrm{~s}\). What is the smallest uncertainty in its decay energy?
(b) Compare this with the rest energy of an electron.
56. The decay energy of a short-lived particle has an uncertainty of 1.0 MeV due to its short lifetime. What is the smallest lifetime it can have?
Solution
\(3.3 \times 10^{-22} \mathrm{~s}\)
57. The decay energy of a short-lived nuclear excited state has an uncertainty of 2.0 eV due to its short lifetime. What is the smallest lifetime it can have?
58. What is the approximate uncertainty in the mass of a muon, as determined from its decay lifetime?
Solution
\(2.66 \times 10^{-46} \mathrm{~kg}\)
59. Derive the approximate form of Heisenberg’s uncertainty principle for energy and time, \(\Delta E \Delta t \approx h\), using the following arguments: Since the position of a particle is uncertain by \(\Delta x \approx \lambda\), where \(\lambda\) is the wavelength of the photon used to examine it, there is an uncertainty in the time the photon takes to traverse \(\Delta x\). Furthermore, the photon has an energy related to its wavelength, and it can transfer some or all of this energy to the object being examined. Thus the uncertainty in the energy of the object is also related to \(\lambda\). Find \(\Delta t\) and \(\Delta E\); then multiply them to give the approximate uncertainty principle.
13.5: Discovery of the Atomic Nucleus
60. Rutherford found the size of the nucleus to be about \(10^{-15} \mathrm{~m}\). This implied a huge density. What would this density be for gold?
Solution
\(6 \times 10^{20} \mathrm{~kg} / \mathrm{m}^{3}\)
61. In Millikan’s oil-drop experiment, one looks at a small oil drop held motionless between two plates. Take the voltage between the plates to be 2033 V, and the plate separation to be 2.00 cm. The oil drop (of density \(0.81 \mathrm{~g} / \mathrm{cm}^{3}\)) has a diameter of \(4.0 \times 10^{-6} \mathrm{~m}\). Find the charge on the drop, in terms of electron units.
62. (a) An aspiring physicist wants to build a scale model of a hydrogen atom for her science fair project. If the atom is 1.00 m in diameter, how big should she try to make the nucleus?
(b) How easy will this be to do?
Solution
(a) \(10.0 ~\mu \mathrm{m}\)
(b) It isn’t hard to make one of approximately this size. It would be harder to make it exactly \(10.0 ~\mu \mathrm{m}\).
13.6: Bohr’s Theory of the Hydrogen Atom
63. By calculating its wavelength, show that the first line in the Lyman series is UV radiation.
Solution
\(\frac{1}{\lambda}=R\left(\frac{1}{n_{\mathrm{f}}^{2}}-\frac{1}{n_{\mathrm{i}}^{2}}\right) \Rightarrow \lambda=\frac{1}{R}\left[\frac{\left(n_{\mathrm{i}} \cdot n_{\mathrm{f}}\right)^{2}}{n_{\mathrm{i}}^{2}-n_{\mathrm{f}}^{2}}\right]\); \(n_{\mathrm{i}}=2\), \(n_{\mathrm{f}}=1\) so that
\(\lambda=\left(\frac{\mathrm{m}}{1.097 \times 10^{7}}\right)\left[\frac{(2 \times 1)^{2}}{2^{2}-1^{2}}\right]=1.22 \times 10^{-7} \mathrm{~m}=122 \mathrm{~nm}\), which is UV radiation.
64. Find the wavelength of the third line in the Lyman series, and identify the type of EM radiation.
65. Look up the values of the quantities in \(a_{\mathrm{B}}=\frac{h^{2}}{4 \pi^{2} m_{e} k q_{e}^{2}}\), and verify that the Bohr radius \(a_{\mathrm{B}}\) is \(0.529 \times 10^{-10} \mathrm{~m}\).
Solution
\(a_{\mathrm{B}}=\frac{h^{2}}{4 \pi^{2} m_{e} k Z q_{e}^{2}}=\frac{\left(6.626 \times 10^{-34} \mathrm{~J} \cdot \mathrm{s}\right)^{2}}{4 \pi^{2}\left(9.109 \times 10^{-31} \mathrm{~kg}\right)\left(8.988 \times 10^{9} \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}\right)(1)\left(1.602 \times 10^{-19} \mathrm{C}\right)^{2}}=0.529 \times 10^{-10} \mathrm{~m}\)
66. Verify that the ground state energy \(E_{0}\) is 13.6 eV by using \({E}_{0}=\frac{2 \pi^{2} q_{e}^{4} m_{e} k^{2}}{h^{2}}\).
67. If a hydrogen atom has its electron in the \(n=4\) state, how much energy in eV is needed to ionize it?
Solution
0.850 eV
68. A hydrogen atom in an excited state can be ionized with less energy than when it is in its ground state. What is \(n\) for a hydrogen atom if 0.850 eV of energy can ionize it?
69. Find the radius of a hydrogen atom in the \(n=2\) state according to Bohr’s theory.
Solution
\(2.12 \times 10^{-10} \mathrm{~m}\)
70. Show that \((13.6 \mathrm{eV}) / h c=1.097 \times 10^{7} \mathrm{~m}=R\) (Rydberg’s constant), as discussed in the text.
71. What is the smallest-wavelength line in the Balmer series? Is it in the visible part of the spectrum?
Solution
365 nm
It is in the ultraviolet.
72. Show that the entire Paschen series is in the infrared part of the spectrum. To do this, you only need to calculate the shortest wavelength in the series.
73. Do the Balmer and Lyman series overlap? To answer this, calculate the shortest-wavelength Balmer line and the longest-wavelength Lyman line.
Solution
No overlap
365 nm
122 nm
74. (a) Which line in the Balmer series is the first one in the UV part of the spectrum?
(b) How many Balmer series lines are in the visible part of the spectrum?
(c) How many are in the UV?
75. A wavelength of \(4.653 ~\mu \mathrm{m}\) is observed in a hydrogen spectrum for a transition that ends in the \(n_{\mathrm{f}}=5\) level. What was \(n_{\mathrm{i}}\) for the initial level of the electron?
Solution
7
76. A singly ionized helium ion has only one electron and is denoted \(\mathrm{He}^{+}\). What is the ion’s radius in the ground state compared to the Bohr radius of hydrogen atom?
77. A beryllium ion with a single electron (denoted \(\mathrm{Be}^{3+}\)) is in an excited state with radius the same as that of the ground state of hydrogen.
(a) What is \(n\) for the \(\mathrm{Be}^{3+}\) ion?
(b) How much energy in eV is needed to ionize the ion from this excited state?
Solution
(a) 2
(b) 54.4 eV
78. Atoms can be ionized by thermal collisions, such as at the high temperatures found in the solar corona. One such ion is \(\mathrm{C}^{+5}\), a carbon atom with only a single electron.
(a) By what factor are the energies of its hydrogen-like levels greater than those of hydrogen?
(b) What is the wavelength of the first line in this ion’s Paschen series?
(c) What type of EM radiation is this?
79. Verify Equations \(r_{n}=\frac{n^{2}}{Z} a_{\mathrm{B}}\) and \(a_{\mathrm{B}}=\frac{h^{2}}{4 \pi^{2} m_{e} k q_{e}^{2}}=0.529 \times 10^{-10} \mathrm{~m}\) using the approach stated in the text. That is, equate the Coulomb and centripetal forces and then insert an expression for velocity from the condition for angular momentum quantization.
Solution
\(\frac{k Z q_{e}^{2}}{r_{n}^{2}}=\frac{m_{e} V^{2}}{r_{n}}\), so that \(r_{n}=\frac{k Z q_{e}^{2}}{m_{e} V^{2}}=\frac{k Z q_{e}^{2}}{m_{e}} \frac{1}{V^{2}}\). From the equation \(m_{e} v r_{n}=n \frac{h}{2 \pi}\), we can substitute for the velocity, giving: \(r_{n}=\frac{k Z q_{e}^{2}}{m_{e}}\) so that \(r_{n}=\frac{n^{2}}{Z} \frac{h^{2}}{4 \pi^{2} m_{e} k q_{e}^{2}}=\frac{n^{2}}{Z} a_{\mathrm{B}}\), where \(a_{\mathrm{B}}=\frac{h^{2}}{4 \pi^{2} m_{e} k q_{e}^{2}}\).
80. The wavelength of the four Balmer series lines for hydrogen are found to be 410.3, 434.2, 486.3, and 656.5 nm. What average percentage difference is found between these wavelength numbers and those predicted by \(\frac{1}{\lambda}=R\left(\frac{1}{n_{f}^{2}}-\frac{1}{n_{1}^{2}}\right)\)? It is amazing how well a simple formula (disconnected originally from theory) could duplicate this phenomenon.


1.1: Pharmacology Basics Introduction
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Safe medication administration is a vital component of the nursing role. Each day it is common for nurses to make critical decisions regarding the safety, appropriateness, and effectiveness of the medications administered to their patients. Examples of decisions that a nurse might make during patient care include:
In order to make safe medication administration decisions, the nurse must have a strong understanding of pharmacology . Symptom management, physical recovery, and individual well-being can be strongly connected to the use of medications in a patient’s treatment plan. Before a student nurse reviews a medication order, checks a medication administration record, or removes a medication from a dispensing machine, it is important to have a foundational understanding of how medications work within the human body. Let’s take a deeper look at the science of pharmacokinetics.


9.3: Skeletal Muscle
By the end of this section, you will be able to:
- Describe the layers of connective tissues that package skeletal muscle
- Explain how muscles work with tendons to move the body
- Identify and describe the microscopic anatomy of a muscle fiber
- Describe excitation-contraction coupling
The best-known feature of skeletal muscle is its ability to contract and cause movement. Skeletal muscles act not only to produce movement but also to stop movement, such as resisting gravity to maintain posture. Small, constant adjustments of skeletal muscles are needed to hold a body upright or balanced in any position. Muscles also prevent excess movement of the bones and joints, maintaining skeletal stability and preventing structural damage or deformation. Joints can become misaligned or dislocated by excessive or inappropriate forces applied to associated bones; muscles work to keep joints stable. Skeletal muscles are located throughout the body at the openings of internal tracts to control the movement of various substances. These muscles allow functions, such as swallowing, urination, and defecation, to be under voluntary control. Skeletal muscles also protect internal organs (particularly abdominal and pelvic organs) by acting as an external barrier or shield to external trauma and by supporting the weight of the organs.
Skeletal muscles contribute to the maintenance of homeostasis in the body by generating heat. Muscle contraction requires energy, and when ATP is broken down, heat is produced. This heat is very noticeable during exercise, when sustained muscle movement causes body temperature to rise, and in cases of extreme cold, when shivering produces random skeletal muscle contractions to generate heat.
Each skeletal muscle is an organ that consists of various integrated tissues. These tissues include the skeletal muscle fibers, blood vessels, nerve fibers, and connective tissue. Each skeletal muscle has three layers of connective tissue (called “mysia”) that enclose it and provide structure to the muscle as a whole, and also compartmentalize the muscle fibers within the muscle (Figure \(\PageIndex{1}\)). Each muscle is wrapped in a sheath of dense, irregular connective tissue called the epimysium , which allows a muscle to contract and move powerfully while maintaining its structural integrity. The epimysium also separates muscle from other tissues and organs in the area, allowing the muscle to move independently.
Inside each skeletal muscle, muscle fibers are organized into bundles, called a fascicle , by a middle layer of dense irregular connective tissue called the perimysium . This fascicular organization is common in muscles of the limbs; it allows the nervous system to trigger a specific movement of a muscle by activating a subset of muscle fibers within a bundle, or fascicle, of the muscle. Inside each fascicle, each muscle fiber is encased in a thin areolar connective tissue layer of collagen and reticular fibers called the endomysium . The endomysium contains the extracellular fluid and nutrients to support the muscle fiber. These nutrients are supplied via blood capillaries located in the endomysium.
In skeletal muscles that work with tendons to pull on bones, the collagen in the three tissue layers intertwines and integrates with the collagen of a tendon, straightening to form a cord of dense regular connective tissue. At the other end of the tendon, the collagen incorporates into periosteum becoming dense irregular connective tissue once more. Tension created by contraction of the muscle fibers is transferred though the mysia, to the tendon, and then to the periosteum to pull on the bone to move the skeleton. In other places, the mysia may fuse with a broad, tendon-like sheet called an aponeurosis , or to fascia, the connective tissue between skin and bones. The broad sheet of connective tissue in the lower back that the latissimus dorsi muscles (the “lats”) fuse into is an example of an aponeurosis.
Every skeletal muscle is richly supplied by blood vessels for nourishment, oxygen delivery, and waste removal. In addition, every muscle fiber in a skeletal muscle is supplied by the axon branch of a somatic motor neuron, which signals the fiber to contract. Unlike cardiac and smooth muscle, the only way to functionally contract a skeletal muscle is through signaling from the nervous system.
The Neuromuscular Junction
The specialized site of the skeletal muscle where a motor neuron’s axon terminal meets the muscle fiber is called the neuromuscular junction (NMJ) ( Figure \(\PageIndex{2}\)). This is where the muscle fiber first responds to signaling by the motor neuron. The sarcolemma (plasma membrane) in the region of a neuromuscular junction, known as the motor end-plate , contains special transmembrane proteins that receive a signal from the motor neuron and begin the electrical signal in the muscle fiber (Figure \(\PageIndex{5}\)). Every skeletal muscle fiber in every skeletal muscle is innervated by a motor neuron at the NMJ. Excitation signals from the neuron are the only way to functionally activate the fiber to contract.
Skeletal Muscle Fibers
Because skeletal muscle cells are long and cylindrical, they are commonly referred to as skeletal muscle fibers . Specific terminology associated with muscle fibers is rooted in the Greek sarco , which means “flesh.” The plasma membrane of muscle fibers is called the sarcolemma and the cytoplasm is referred to as sarcoplasm. Skeletal muscle fibers can be quite large for human cells, with diameters up to 100 μ m and lengths up to 30 cm (11.8 in). During early development, embryonic myoblasts, each with its own nucleus, fuse with up to hundreds of other myoblasts to form the multinucleated skeletal muscle fibers. Multiple nuclei mean multiple copies of genes, permitting the production of the large amounts of proteins and enzymes needed for muscle contraction along the length of the entire muscle fiber.
Within the sarcoplasm of each muscle fiber are myofibrils that run the entire length of the muscle fiber and attach to the sarcolemma at its end (Figure \(\PageIndex{1}\)). Each myofibril is composed of a bundle of contractile myofilaments (known as actin and myosin), along with other support proteins. As myofibrils contract, the entire muscle cell contracts. Because myofibrils are only approximately 1.2 μ m in diameter, hundreds to thousands can be found inside one muscle fiber. Surrounding each myofibril is the sarcoplasmic reticulum (SR), the specialized smooth endoplasmic reticulum of muscle fibers, which stores, releases, and retrieves calcium ions (Ca ++ ) (Figure \(\PageIndex{3}\)).
As will be discussed, the signal from a motor neuron actually stimulates the release of calcium ions (Ca ++ ) from its storage in the cell’s SR. For the action potential to reach the membrane of the SR from the motor end plate, there are periodic invaginations in the sarcolemma, called T-tubules (“T” stands for “transverse”). You will recall that the diameter of a muscle fiber can be up to 100 μ m, so these T-tubules ensure that the membrane can get close to the SR in the sarcoplasm. On each side of the T-tubule is a terminal cisterna (plural - cisternae), an enlarged region of the SR that contains high concentrations of calcium ions. The arrangement of a T-tubule with the terminal cisternae of the SR on either side is called a triad (Figure \(\PageIndex{3}\)). The triad surrounds the cylindrical structure of the myofibril.
The Sarcomere
The striated appearance of skeletal muscle fibers is due to the arrangement of the myofilaments actin and myosin in sequential order from one end of the muscle fiber to the other. Each segment of these myofilaments and their regulatory proteins, troponin and tropomyosin (along with other proteins), is called a sarcomere . The sarcomere is the functional unit of the muscle fiber. A chain of thousands of repeating sarcomeres forms the long tube-like myofibrils.
Each sarcomere is approximately 2 μ m in length with a three-dimensional cylinder-like arrangement. The repeating pattern of the myofilaments, along with other proteins, within the sarcomere gives rise to very distinctive regions and structures (Figure \(\PageIndex{4}\)). Each sarcomere is bordered by structures called Z-discs (also called Z-lines, because pictures are two-dimensional), to which the actin myofilaments are anchored. Because the actin forms strands that are thinner than the myosin, it is called the thin filament of the sarcomere. Likewise, because the myosin strands have more mass and are thicker, they are called the thick filament of the sarcomere. This difference in thickness, is what causes the striated appearance of skeletal muscle fibers. The area immediately surrounding the Z-discs, is known as the I band , or light band, because only actin is present. The thin composition of actin makes it difficult to see, hence the name the light band. The myosin is anchored in the center of the sarcomere, to a structure known as the M line, where is projects toward but not all to way to, the Z-discs. The area of the sarcomere where myosin is present is known as the A band , or dark band. Opposite to thin composition of actin, myosin is thick, which makes this area dark in appearance. There are some areas, where actin and myosin myofilaments are both present, known as the zone of overlap. Immediately surrounding the M line is the H zone, where there is no overlap and only myosin is present.
Myofilaments
As mentioned above, actin is known as the thin filament and myosin as the thick filament. The structural differences between these two myofilaments not only leads to the striated appearance of skeletal muscle tissue, but also to the ability of the two to work together to produce the overall function of muscle contraction.
Actin myofilaments are formed by multiple actin subunits, each of which contain a myosin binding site (Figure \(\PageIndex{5}\)). In a relaxed muscle, the binding sites remain shielded by the troponin-tropomyosin regulatory complex. The tropomyosin filaments run the length of the actin, covering all of the binding sites and are held in place by troponin .
Myosin myofilaments have two specific regions, the tail and the heads (Figure \(\PageIndex{5}\)). Connecting these two regions is a flexible hinge region. The tails function to cluster many myosin myofilaments together, while the heads take an active role in the muscle contraction process. The heads contain an actin-binding site, as well as an ATP binding site. The importance of these binding sites will be discussed in the next section of this book.
Excitation-Contraction Coupling
All living cells have membrane potentials, or electrical gradients across their membranes. The inside of the membrane is usually around -60 to -90 mV, relative to the outside. This is referred to as a cell’s membrane potential. Neurons and muscle cells can use their membrane potentials to generate electrical signals. They do this by controlling the movement of charged particles, called ions, across their membranes to create electrical currents. This is achieved by opening and closing specialized proteins in the membrane called ion channels. Although the currents generated by ions moving through these channel proteins are very small, they form the basis of both neural signaling and muscle contraction.
Both neurons and skeletal muscle cells are electrically excitable, meaning that they are able to generate action potentials. An action potential is a special type of electrical signal that can travel along a cell membrane as a wave. This allows a signal to be transmitted quickly and faithfully over long distances.
Although the term excitation-contraction coupling confuses or scares some students, it comes down to this: for a skeletal muscle fiber to contract, its membrane must first be “excited”—in other words, it must be stimulated to fire an action potential. The muscle fiber action potential, which sweeps along the sarcolemma as a wave, is “coupled” to the actual contraction through the release of calcium ions (Ca ++ ) from the SR. Once released, the Ca ++ interacts with the shielding proteins, forcing them to move aside so that the actin-binding sites are available for attachment by myosin heads. The myosin then pulls the actin filaments toward the center, shortening the muscle fiber.
In skeletal muscle, this sequence begins with signals from the somatic motor division of the nervous system. In other words, the “excitation” step in skeletal muscles is always triggered by signaling from the nervous system (Figure \(\PageIndex{6}\)).
The motor neurons that tell the skeletal muscle fibers to contract originate in the spinal cord, with a smaller number located in the brainstem for activation of skeletal muscles of the face, head, and neck. These neurons have long processes, called axons, which are specialized to transmit action potentials long distances— in this case, all the way from the spinal cord to the muscle itself (which may be up to three feet away). The axons of multiple neurons bundle together to form nerves, like wires bundled together in a cable.
Signaling begins when a neuronal action potential travels along the axon of a motor neuron, and then along the individual branches to terminate at the NMJ. At the NMJ, the axon terminal releases a chemical messenger, or neurotransmitter , called acetylcholine (ACh) . The ACh molecules diffuse across a minute space called the synaptic cleft and bind to ACh receptors located within the motor end-plate of the sarcolemma on the other side of the synapse. Once ACh binds, a channel in the ACh receptor opens and positively charged ions can pass through into the muscle fiber, causing it to depolarize , meaning that the membrane potential of the muscle fiber becomes less negative (closer to zero.)
As the membrane depolarizes, another set of ion channels called voltage-gated sodium channels are triggered to open. Sodium ions enter the muscle fiber, and an action potential rapidly spreads (or “fires”) along the entire membrane to initiate excitation-contraction coupling.
Things happen very quickly in the world of excitable membranes (just think about how quickly you can snap your fingers as soon as you decide to do it). Immediately following depolarization of the membrane, it repolarizes, re-establishing the negative membrane potential. Meanwhile, the ACh in the synaptic cleft is degraded by the enzyme acetylcholinesterase (AChE) so that the ACh cannot rebind to a receptor and reopen its channel, which would cause unwanted extended muscle excitation and contraction. Propagation of an action potential along the sarcolemma is the excitation portion of excitation-contraction coupling. The T-tubules carry the action potential into the interior of the cell, which triggers the opening of calcium channels in the membrane of the adjacent SR, causing Ca ++ to diffuse out of the SR and into the sarcoplasm. It is the arrival of Ca ++ in the sarcoplasm that initiates contraction of the muscle fiber by its contractile units, or sarcomeres.
Concept Review
Skeletal muscles contain connective tissue, blood vessels, and nerves. There are three layers of connective tissue: epimysium, perimysium, and endomysium. Skeletal muscle fibers are organized into groups called fascicles. Blood vessels and nerves enter the connective tissue and branch. Muscles attach to bones directly or through tendons or aponeuroses. Skeletal muscles maintain posture, stabilize joints, support organs, control internal movement, and generate heat.
Skeletal muscle fibers are long, multinucleated cells. The membrane of the cell is the sarcolemma; the cytoplasm of the cell is the sarcoplasm. The sarcoplasmic reticulum (SR) is a form of endoplasmic reticulum that specializes in storing and releasing calcium. Muscle fibers are composed of myofibrils. The banding pattern of myofibrils, called striations, are created by the organization of actin and myosin proteins in functional units called sarcomeres.
Review Questions
Q. The correct order for the smallest to the largest unit of organization in muscle tissue is ________.
A. fascicle, myofilament, muscle fiber, myofibril
B. myofilament, myofibril, muscle fiber, fascicle
C. muscle fiber, fascicle, myofilament, myofibril
D. myofibril, muscle fiber, myofilament, fascicle
- Answer
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Answer: B
Q. Depolarization of the sarcolemma means ________.
A. the inside of the membrane has become less negative as sodium ions accumulate
B. the outside of the membrane has become less negative as sodium ions accumulate
C. the inside of the membrane has become more negative as sodium ions accumulate
D. the sarcolemma has completely lost any electrical charge
- Answer
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Answer: A
Critical Thinking Questions
Q. What would happen to skeletal muscle if the epimysium were destroyed?
- Answer
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A. Muscles would lose their integrity during powerful movements, resulting in muscle damage.
Q. Describe how tendons facilitate body movement.
- Answer
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A. When a muscle contracts, the force of movement is transmitted through the tendon, which pulls on the bone to produce skeletal movement.
Q. What are the five primary functions of skeletal muscle?
- Answer
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A. Produce movement of the skeleton, maintain posture and body position, support soft tissues, encircle openings of the digestive, urinary, and other tracts, and maintain body temperature.
Q. What are the opposite roles of voltage-gated sodium channels and voltage-gated potassium channels?
- Answer
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A. The opening of voltage-gated sodium channels, followed by the influx of Na + , transmits an Action Potential after the membrane has sufficiently depolarized. The delayed opening of potassium channels allows K + to exit the cell, to repolarize the membrane.
Glossary
- A band
- region of the sarcomere where myosin is present
- acetylcholine (ACh)
- neurotransmitter that binds at a motor end-plate to trigger depolarization
- actin
- protein that makes up most of the thin myofilaments in a sarcomere muscle fiber
- action potential
- change in voltage of a cell membrane in response to a stimulus that results in transmission of an electrical signal; unique to neurons and muscle fibers
- aponeurosis
- broad, tendon-like sheet of connective tissue that attaches a skeletal muscle to another skeletal muscle or to a bone
- depolarize
- to reduce the voltage difference between the inside and outside of a cell’s plasma membrane (the sarcolemma for a muscle fiber), making the inside less negative than at rest
- endomysium
- loose, and well-hydrated connective tissue covering each muscle fiber in a skeletal muscle
- epimysium
- outer layer of connective tissue around a skeletal muscle
- excitation-contraction coupling
- sequence of events from motor neuron signaling to a skeletal muscle fiber to contraction of the fiber’s sarcomeres
- fascicle
- bundle of muscle fibers within a skeletal muscle
- H zone
- region of the sarcomere surrounding the M line where only myosin is present
- I band
- also known as the light band; region of the sarcomere where only actin is present
- M line
- center structure of a sarcomere that serves to anchor myosin filaments
- motor end-plate
- sarcolemma of muscle fiber at the neuromuscular junction, with receptors for the neurotransmitter acetylcholine
- myofibril
- long, cylindrical groups of myofilaments (actin and myosin) that run parallel within the muscle fiber and contains the sarcomeres
- myofilament
- proteins, actin and myosin, that make up myofibrils
- myosin
- protein that makes up most of the thick cylindrical myofilament within a sarcomere muscle fiber
- neuromuscular junction (NMJ)
- synapse between the axon terminal of a motor neuron and the section of the membrane of a muscle fiber with receptors for the acetylcholine released by the terminal
- neurotransmitter
- signaling chemical released by nerve terminals that bind to and activate receptors on target cells
- perimysium
- connective tissue that bundles skeletal muscle fibers into fascicles within a skeletal muscle
- sarcomere
- longitudinally, repeating functional unit of skeletal muscle, with all of the contractile and associated proteins involved in contraction
- sarcolemma
- plasma membrane of a skeletal muscle fiber
- sarcoplasm
- cytoplasm of a muscle cell
- sarcoplasmic reticulum (SR)
- specialized smooth endoplasmic reticulum, which stores, releases, and retrieves Ca ++
- skeletal muscle fiber
- multinucleated skeletal muscle cells
- synaptic cleft
- space between a nerve (axon) terminal and a motor end-plate
- T-tubule
- projection of the sarcolemma into the interior of the cell
- terminal cisternae
- enlarged regions of the sarcoplamic reticulum found on either side of the t-tubules
- thick filament
- the thick myosin strands and their multiple heads projecting from the center of the sarcomere toward, but not all to way to, the Z-discs
- thin filament
- thin strands of actin and its troponin-tropomyosin complex projecting from the Z-discs toward the center of the sarcomere
- triad
- the grouping of one T-tubule and two terminal cisternae
- troponin
- regulatory protein that binds to actin, tropomyosin, and calcium
- tropomyosin
- regulatory protein that covers myosin-binding sites to prevent actin from binding to myosin
- voltage-gated sodium channels
- membrane proteins that open sodium channels in response to a sufficient voltage change, and initiate and transmit the action potential as Na + enters through the channel
- Z-disc
- also Z-line; structure that forms the border of a sarcomere and serves as the anchor site for actin.
Contributors and Attributions
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OpenStax Anatomy & Physiology (CC BY 4.0). Access for free at https://openstax.org/books/anatomy-and-physiology


2.1: SDLC
Systems-Development Life Cycle
The first development methodology we are going to review is the Systems Development Life Cycle (SDLC) . This methodology was first developed in the 1960s to manage the large software projects associated with corporate systems running on mainframes. It is a very structured and risk-averse methodology designed to manage large projects that included multiple programmers and systems that would have a large impact on the organization.
SDLC waterfall (click to enlarge)
Various definitions of the SDLC methodology exist, but most contain the following phases.
- Preliminary Analysis . In this phase, a review is done of the request. Is creating a solution possible? What alternatives exist? What is currently being done about it? Is this project a good fit for our organization? A key part of this step is a feasibility analysis, which includes an analysis of the technical feasibility (is it possible to create this?), the economic feasibility (can we afford to do this?), and the legal feasibility (are we allowed to do this?). This step is important in determining if the project should even get started.
- System Analysis. In this phase, one or more system analysts work with different stakeholder groups to determine the specific requirements for the new system. No programming is done in this step. Instead, procedures are documented, key players are interviewed, and data requirements are developed in order to get an overall picture of exactly what the system is supposed to do. The result of this phase is a system-requirements document.
- System Design. In this phase, a designer takes the system-requirements document created in the previous phase and develops the specific technical details required for the system. It is in this phase that the business requirements are translated into specific technical requirements. The design for the user interface, database, data inputs and outputs, and reporting are developed here. The result of this phase is a system-design document. This document will have everything a programmer will need to actually create the system.
- Development. The system becomes further established in the development phase. Using the system-design document as a guide, a programmer (or team of programmers) develops the program. The result of this phase is an initial working program that meets the requirements laid out in the system-analysis phase and the design developed in the system-design phase.
- Testing . In the testing phase, the software program or system developed in the previous phase is put through a series of structured tests. The first is a unit test, which tests individual parts of the code or system for errors or bugs. Next is a system test, where the different components of the system are tested to ensure that they work together properly. Finally, the user-acceptance test allows those that will be using the software to test the system to ensure that it meets their standards. Any bugs, errors, or problems found during testing are addressed and then tested again.
- Implementation. Once the new system is developed and tested, it has to be implemented in the organization. This phase includes training the users, providing documentation, and converting from any previous system to the new system. Implementation can take many forms, depending on the type of system, the number and type of users, and how urgent it is that the system becomes operational. These different forms of implementation are covered later in the chapter.
- Maintenance. This final phase takes place once the implementation phase is complete. In this phase, the system has a structured support process in place: reported bugs are fixed and requests for new features are evaluated and implemented; system updates and backups are performed on a regular basis.
The SDLC methodology is sometimes referred to as the waterfall methodology to represent how each step is a separate part of the process; only when one step is completed can another step begin. After each step, an organization must decide whether to move to the next step or not. This methodology has been criticized for being quite rigid. For example, changes to the requirements are not allowed once the process has begun. No software is available until after the programming phase.
Again, SDLC was developed for large, structured projects. Projects using SDLC can sometimes take months or years to complete. Because of its inflexibility and the availability of new programming techniques and tools, many other software-development methodologies have been developed. Many of these retain some of the underlying concepts of SDLC but are not as rigid.


16.1: Nervous System Introduction
Learning Objectives
- Apply the rules of medical language to build, analyze, spell, pronounce, abbreviate, and define terms as they relate to the nervous system
- Identify meanings of key word components of the nervous system
- Categorize diagnostic, therapeutic, procedural, or anatomic terms related to the nervous system
- Use terms related to the nervous system
- Use terms related to the diseases and disorders of the nervous system
Introduction to the Nervous System
The nervous system includes the brain, the spinal cord, and a complex system of nerves. It uses electrical signals and chemical substances called neurotransmitters for communication across body parts.
This chapter will review common word components related to the nervous system to assist learners in analyzing, building, and defining medical terms. Other terms, whose definitions cannot be easily built from word components, will be described in context based on the anatomy and physiology of the nervous system and common diseases and disorders. Medical specialists, diagnostic tests, and procedures related to the nervous system will also be discussed.
View a supplementary YouTube video [1] on the nervous system by Crash Course: The Nervous System, Part 1: Crash Course Anatomy & Physiology #8
- CrashCourse. (2015, February 23). The nervous system, Part 1: Crash Course Anatomy & Physiology #8 [Video]. YouTube. All rights reserved. https://www.youtube.com/watch?v=qPix_X-9t7E ↵


26.6: Amino Acid Analysis of Peptides
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- Last updated
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- Steven Farmer, Dietmar Kennepohl, & R Balaji Rao
- University of Illinois Springfield
Objectives
After completing this section, you should be able to describe, briefly, how the identity and amounts of each amino acid residue present in a peptide of unknown structure may be determined.
Key Terms
Make certain that you can define, and use in context, the key term below.
- amino acid analyzer
Study Notes
You need not memorize the reaction between ninhydrin and an α ‑amino acid.
Ion-exchange chromatography
When a protein is to be analysed, it is first heated with acid to hydrolyse all the peptide bonds. When such a mixture of amino acids is to be purified and estimated quantitatively, ion-exchange chromatography is the technique of choice. Fully automated amino acid analyzers are now available, which are equipped with a solvent pump to deliver the required buffer(s) in a programmed manner. There is a column, filled with Dowex 50 resin (Fig 26.5.1). This solid support is made up of polymeric beads. Chemically speaking they are polymers bearing arylsulfonic acid groups. The cation exchange resin helps in the separation of amino acids. In a typical run (Fig 26.5.2), the eluent is a buffer. The pH value of the buffer could be varied as step elution or as gradient elution. The chromatogram shown in Fig 26.5.2 is a chromatogram run with gradient elution technique, using ninhydrin as the post column treatment. The detector is a UV detector scanning the wavelengths 570 nm and 440 nm.
Fig 26.5.1: A Cation Rasin like Dowex 50 is a polymeric bead bearing aryl sulfonic acid groups
Fig 26.5.2: Some typical chromatograms from an amino acid analyzer
The Ninhydrin Reaction
Alpha-amino acids show reactivity at their the carboxylic acid and amine sites typical of those functional groups. In addition to these common reactions of amines and carboxylic acids, common alpha-amino acids, except proline, undergo a unique reaction with the triketohydrindene hydrate known as ninhydrin. Among the products of this unusual reaction (shown on the left below) is a purple colored imino derivative, which provides as a useful color test for these amino acids, most of which are colorless. A common application of the ninhydrin test is the visualization of amino acids in paper chromatography. As shown in the graphic on the right, samples of amino acids or mixtures thereof are applied along a line near the bottom of a rectangular sheet of paper (the baseline). The bottom edge of the paper is immersed in an aqueous buffer, and this liquid climbs slowly toward the top edge. As the solvent front passes the sample spots, the compounds in each sample are carried along at a rate which is characteristic of their functionality, size and interaction with the cellulose matrix of the paper. Some compounds move rapidly up the paper, while others may scarcely move at all. The ratio of the distance a compound moves from the baseline to the distance of the solvent front from the baseline is defined as the retardation (or retention) factor R f . Different amino acids usually have different R f 's under suitable conditions. In the example on the right, the three sample compounds (1, 2 & 3) have respective R f values of 0.54, 0.36 & 0.78.


13.2: The global sulfur cycle
Key Points
- Sulfur is an essential element for the macromolecules of living things since it determines the 3-D folding patterns of proteins.
- On land, sulfur enters the atmosphere via acid rain, fallout, the weathering of rocks, decompostion of organic materials, and geothermal vents.
- Sulfur enters the ocean via runoff, fallout, and underwater geothermal vents; some marine ecosystems also rely on chemoautotrophs as a sulfur source.
- The burning of fossil fuels increases the amount of sulfide in the atmosphere and causes acid rain.
- Acid rain is corrosive rain that causes damage to aquatic ecosystems by lowering the pH of lakes, killing many of the resident fauna; it also degrades buildings and human-made structures.
The Sulfur Cycle
Sulfur is an essential element for the macromolecules of living things. As a part of the amino acid cysteine, it is involved in the formation of disulfide bonds within proteins, which help to determine their 3-D folding patterns and, hence, their functions. As shown in Figure \(\PageIndex{1}\), sulfur cycles between the oceans, land, and atmosphere. Atmospheric sulfur is found in the form of sulfur dioxide (SO 2 ), which enters the atmosphere in three ways: first, from the decomposition of organic molecules; second, from volcanic activity and geothermal vents; and, third, from the burning of fossil fuels by humans.
Figure \(\PageIndex{1}\): The sulfur cycle. Sulfur dioxide (SO 2 ) from the atmosphere is dissolved in precipitation as weak sulfuric acid or falls directly to Earth as fallout. This releases sulfates (SO 4 2 - ) into the soil and water. Soil sulfates can be carried as runoff into the water. Marine sulfate can form pyrite, and this can break down to release soil sulfates. Organisms in terrestrial and marine ecosystems assimilate sulfate, adding sulfur to organic molecules, such as proteins (not shown). Decomposition of these organisms returns sulfates to the soil. Microorganisms can convert sulfates to hydrogen sulfide (H 2 S) and vice versa. Decomposition, volcanic eruptions, and human activities (including burning fossil fuels) can release hydrogen sulfide (H 2 S) or sulfur dioxide into the atmosphere. (credit: Melissa Ha modification of work by John M. Evans and Howard Perlman, USGS)On land, sulfur is deposited in five major ways: precipitation, direct fallout from the atmosphere, rock weathering, decomposition of organic material and geothermal vents (Figure \(\PageIndex{2}\)). As rain falls through the atmosphere, sulfur dioxide (SO 2 ) in the atmosphere is dissolved in the form of weak sulfuric acid (H 2 SO 4 ), creating acid rain. Sulfur can also fall directly from the atmosphere in a process called fallout. The weathering of sulfur-containing rocks also releases sulfur into the soil. These rocks originate from ocean sediments that are moved to land by the geologic uplift. Terrestrial ecosystems can then make use of these soil sulfates (SO 4 2 - ), which enter the food web by being taken up by plant roots. When these plants decompose and die, sulfur is released back into the atmosphere as hydrogen sulfide (H 2 S) gas.
Figure \(\PageIndex{2}\): At this sulfur vent in Lassen Volcanic National Park in northeastern California, the yellowish sulfur deposits are visible near the mouth of the vent. (credit: “Calbear22”/Wikimedia Commons)Sulfur enters the ocean in runoff from land, from atmospheric fallout, and from hydrothermal vents. Some ecosystems rely on microorganisms using sulfur as a biological energy source (chemoautotrophs) in contrast to ecosystems with photosynthetic producers. This sulfur then supports marine ecosystems in the form of sulfates.
Human activities have played a major role in altering the balance of the global sulfur cycle. The burning of large quantities of fossil fuels, especially from coal, releases sulfur dioxide, which reacts with the atmosphere to form sulfuric acid. Like nitric acid, sulfuric acid contributes to acid deposition. Acid deposition (sometimes referred to simply as acid rain) damages the natural environment by lowering the pH of lakes, thus killing many of the resident plants and animals. Acid deposition also affects the man-made environment through the chemical degradation of buildings. For example, many marble monuments, such as the Lincoln Memorial in Washington, DC, have suffered significant damage from acid rain over the years. These examples show the wide-ranging effects of human activities on our environment and the challenges that remain for our future.
Chemosynthesis
Why do bacteria that live deep below the ocean’s surface rely on chemical compounds instead of sunlight for energy to make food?
Most autotrophs make food by photosynthesis, but this isn’t the only way that autotrophs produce food. Some bacteria make food by another process, which uses chemical energy instead of light energy. This process is called chemosynthesis . In chemosynthesis, one or more carbon molecules (usually carbon dioxide or methane, CH 4 ) and nutrients is converted into organic matter, using the oxidation of inorganic molecules (such as hydrogen gas, hydrogen sulfide (H 2 S) or ammonia (NH 3 )) or methane as a source of energy, rather than sunlight. In hydrogen sulfide chemosynthesis, in the presence of carbon dioxide and oxygen, carbohydrates (CH 2 O) can be produced:
CO 2 + O 2 + 4H 2 S → CH 2 O + 4S + 3H 2 O
Many organisms that use chemosynthesis are extremophiles, living in harsh conditions, such as in the absence of sunlight and a wide range of water temperatures, some approaching the boiling point. Some chemosynthetic bacteria live around deep-ocean vents known as “black smokers.” Compounds such as hydrogen sulfide, which flow out of the vents from Earth’s interior, are used by the bacteria for energy to make food. These organisms are known as chemoautotrophs. Many chemosynthetic microorganisms are consumed by other organisms in the ocean, and symbiotic associations between these organisms and respiring heterotrophs are quite common. Consumers that depend on these bacteria to produce food for them include giant tubeworms (Figure \(\PageIndex{3}\)), and Pompeii worms ((Figure \(\PageIndex{4}\)).
Figure \(\PageIndex{3}\): Tubeworms deep in the Galapagos Rift get their energy from chemosynthetic bacteria.  Their survival depends on a symbiotic relationship with the billions of bacteria that live inside them. These bacteria convert the chemicals that shoot out of the hydrothermal vents into food for the worm.
Figure \(\PageIndex{3}\): Is it possible to live in temperatures over 175°F?  The Pompeii worm, the most heat -tolerant animal on Earth, lives in the deep ocean at super-heated hydrothermal vents. Covering this deep-sea worm's back is a fleece of bacteria. These microbes contain all the genes necessary for life in extreme environments.
Contributors and Attributions
This section was modified from the original by Kyle Whittinghill
-
Samantha Fowler (Clayton State University), Rebecca Roush (Sandhills Community College), James Wise (Hampton University). Original content by OpenStax (CC BY 4.0; Access for free at https://cnx.org/contents/b3c1e1d2-83...4-e119a8aafbdd ).
- 20.2: Biogeochemical Cycles by OpenStax , is licensed CC BY
- 2.24: Chemosynthesis by CK-12: Biology Concepts , is licensed CC BY-NC
- General Microbiology Provided by : Boundless.com. License : CC BY-SA: Attribution-ShareAlike
- Atmosphere-Biosphere-Hydrosphere-Lithosphere. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi...ithosphere.png . License : CC BY: Attribution